Primitive trinomials over GF(2)
The form of these primitive polynomials is P(x)=xd+xn+1
with n < d, and d a Mersenne number. This last property ensures that 2d-1 is a prime number, and therefore all irreducible polynomials of degree d are primitive.
These irreducible polynomials can be displayed in Maple
format to allow easy check.
Mersenne degrees
These tables are due to Lee D. Killough
(killough@bach.convex.com).
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Index:
2
3
5
7
13
17
19
31
61
89
107
127
521
607
1279
2203
2281
3217
4423
9689
19937
1+X+X2
There is 1 primitive trinomial of degree 2
1+X+X3
1+X2+X3
There are 2 primitive trinomials of degree 3
1+X2+X5
1+X3+X5
There are 2 primitive trinomials of degree 5
1+X+X7
1+X6+X7
1+X3+X7
1+X4+X7
There are 4 primitive trinomials of degree 7
There is no primitive trinomial of degree 13
1+X3+X17
1+X14+X17
1+X5+X17
1+X12+X17
1+X6+X17
1+X11+X17
There are 6 primitive trinomials of degree 17
There is no primitive trinomial of degree 19
1+X3+X31
1+X28+X31
1+X6+X31
1+X25+X31
1+X7+X31
1+X24+X31
1+X13+X31
1+X18+X31
There are 8 primitive trinomials of degree 31
There is no primitive trinomial of degree 61
1+X38+X89
1+X51+X89
There are 2 primitive trinomials of degree 89
There is no primitive trinomial of degree 107
1+X+X127
1+X126+X127
1+X7+X127
1+X120+X127
1+X15+X127
1+X112+X127
1+X30+X127
1+X97+X127
1+X63+X127
1+X64+X127
There are 10 primitive trinomials of degree 127
1+X32+X521
1+X489+X521
1+X48+X521
1+X473+X521
1+X158+X521
1+X363+X521
1+X168+X521
1+X353+X521
There are 8 primitive trinomials of degree 521
1+X105+X607
1+X502+X607
1+X147+X607
1+X460+X607
1+X273+X607
1+X334+X607
There are 6 primitive trinomials of degree 607
1+X216+X1279
1+X1063+X1279
1+X418+X1279
1+X861+X1279
There are 4 primitive trinomials of degree 1279
There is no primitive trinomial of degree 2203
1+X715+X2281
1+X915+X2281
1+X1029+X2281
1+X1252+X2281
1+X1366+X2281
1+X1566+X2281
There are 6 primitive trinomials of degree 2281
1+X67+X3217
1+X3150+X3217
1+X576+X3217
1+X2641+X3217
There are 4 primitive trinomials of degree 3217
1+X271+X4423
1+X4152+X4423
1+X369+X4423
1+X4054+X4423
1+X370+X4423
1+X4053+X4423
1+X649+X4423
1+X3774+X4423
1+X1393+X4423
1+X3030+X4423
1+X1419+X4423
1+X3004+X4423
1+X2098+X4423
1+X2325+X4423
There are 14 primitive trinomials of degree 4423
1+X84+X9689
1+X9605+X9689
1+X471+X9689
1+X9218+X9689
1+X1836+X9689
1+X7853+X9689
1+X2444+X9689
1+X7245+X9689
1+X4187+X9689
1+X5502+X9689
There are 10 primitive trinomials of degree 9689
1+X881+X19937
1+X19056+X19937
1+X7083+X19937
1+X12854+X19937
1+X9842+X19937
1+X10095+X19937
There are 6 primitive trinomials of degree 19937
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